For what value of the $c$ , if the root of $(c - 2) x^{2} + 2 (c - 2) x + 2 = 0$ are not real.

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Solution

Given quadratic equation :

$(c - 2) x^{2} + 2 (c - 2) x + 2 = 0$

For imaginary roots,

$D < 0$ in $a x^{2} + b x + c = 0$

$b^{2} - 4 a c < 0$

Hence,

$\Rightarrow$ $2^{2} (c - 2)^{2} - 4 (c - 2) (2) < 0$

$\Rightarrow$ $4 (c - 2)^{2} - 8 (c - 2) < 0$

$\Rightarrow$ $(c - 2) [4 (- 2) - 8] < 0$

$\Rightarrow$ $4 (c - 2) [c - 4] < 0$

$\Rightarrow$ $4 (c - 2) (c - 4) < 0$

$c \in (2, 4)$

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